A graphical approach to understanding the form of the centripetal acceleration.
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Assumptions
We assume that we have uniform circular motion (motion with a constant radius and a constant speed centered at a fixed point in space).
The Diagram
The picture below illustrates the motion, with coordinates chosen so that the angular position at t = 0 is θ = 0.
To the right of the motion diagram is a vector diagram that shows the change in the velocity vector. The picture motivates the conclusion that if we take a very small Δt, the change in the velocity approaches:
\begin
[ \Delta\vec
\rightarrow - v(\Delta \theta)\hat
]\end
In the infinitesimal limit, this equation becomes:
\begin
[ \frac{d\vec{v}}
= - v \frac
\hat
]\end
Using the fact that for uniform circular motion,
\begin
[ \frac
= \frac
]\end
we arrive at the form of the centripetal acceleration:
\begin
[ \vec
= \frac{d\vec{v}}
= -\frac{v^{2}}
\hat
]\end
Analogy with Gyroscopic Precession
Consider a gyroscope precessing. The angular momentum will trace out a circle as shown below.